# Counting Subgroups in a Family of Nilpotent Semi‐Direct Products

@article{Voll2004CountingSI, title={Counting Subgroups in a Family of Nilpotent Semi‐Direct Products}, author={Christopher Voll}, journal={Bulletin of the London Mathematical Society}, year={2004}, volume={38} }

In this paper we compute the subgroup zeta functions of nilpotent groups of the form Gn:=〈x1,…,xn,y1,…,yn−1∣[xi,xn]=yi,1⩽i⩽n−1 , all other [,] trivial 〉 and deduce local functional equations. 2000 Mathematics Subject Classification 11M41, 20E07.

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## References

SHOWING 1-10 OF 21 REFERENCES

### Normal subgroup growth in free class-2-nilpotent groups

- Mathematics
- 2004

Abstract.Let F2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions prove that they satisfy local functional equations and determine their abscissae of…

### Analytic properties of zeta functions and subgroup growth

- Mathematics
- 2000

It has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group G. Although the story began much…

### Functional equations and uniformity for local zeta functions of nilpotent groups

- Mathematics
- 1996

We investigate in this paper the zeta function [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/] associated to a nilpotent group Γ introduced in [GSS]. This zeta function…

### Zeta functions of groups: zeros and friendly ghosts

- Mathematics
- 2002

<abstract abstract-type="TeX"><p>We consider zeta functions defined as Euler products of <i>W</i>(<i>p, p<sup>-s</sup></i>) over all primes <i>p</i> where <i>W</i>(<i>X, Y</i>) is a polynomial. Such…

### Functional equations for local normal zeta functions of nilpotent groups

- Mathematics
- 2003

Abstract.We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically…

### Groups St Andrews 2001 in Oxford: Zeta functions of groups: the quest for order versus the flight from ennui

- Mathematics
- 2003

### Proisomorphic zeta functions of groups

- PhD Thesis, University of Oxford,
- 2005